srm {nnR}

srm

Description

The function that returns the right scalar multiplication neural network

Usage

srm(nu, a)

nu %srm% a

Arguments

nu

A neural network

a

A real number.

Value

Returns a neural network that is $\nu \triangleleft a$. This instantiates as $f(a \cdot x)$.under continuous function activation. More specifically we will define this operation as:

Let $\lambda \in \mathbb{R}$. We will denote by $(\cdot) \triangleleft (\cdot): \mathsf{NN} \times \mathbb{R} \rightarrow \mathsf{NN}$ the function satisfying for all $\nu \in \mathsf{NN}$ and $\lambda \in \mathbb{R}$ that $\nu \triangleleft \lambda = \nu \bullet \mathsf{Aff}_{\lambda \mathbb{I}_{\mathsf{I}(\nu)},0}$.

References

Definition 2.3.4. Jentzen, A., Kuckuck, B., and von Wurstemberger, P. (2023). Mathematical introduction to deep learning: Methods, implementations, and theory. https://arxiv.org/abs/2310.20360.

Note: We will have two versions of this operation, a prefix and an infix version.